To set up the equation based on the problem, we know that angles BCD and DCE are complementary, meaning they add up to 90 degrees. We have:
\[ \angle BCD + \angle DCE = 90^\circ \]
Given that \(\angle BCD = 63^\circ\) and \(\angle DCE = 4.5x^\circ\), we can substitute these values into the equation:
\[ 63^\circ + 4.5x^\circ = 90^\circ \]
Now, we will solve for \(x\):
-
Subtract 63 from both sides: \[ 4.5x = 90 - 63 \] \[ 4.5x = 27 \]
-
Now, divide both sides by 4.5 to find \(x\): \[ x = \frac{27}{4.5} \]
To divide, we can convert 4.5 into a fraction: \[ 4.5 = \frac{9}{2} \] Thus, \[ x = \frac{27}{\frac{9}{2}} = 27 \times \frac{2}{9} = 3 \times 2 = 6 \]
So, the value of \(x\) is:
\[ \boxed{6} \]
Therefore, the answer is \(x = 6\).